Misunderstandings 1 and 2

We first load a few packages, and define a ggplot theme.

# Packages
  library(tidyverse)
  library(igraph)
  library(ggraph)
  library(tidygraph)
  library(cmdstanr)
  library(tidybayes)

# ggplot2 theme
  theme <- theme_set(
    theme_bw() +
      theme(
      panel.grid = element_blank(),
      axis.ticks = element_blank()
      ))

1 Dependencies as a nuisance

Below, we code up simple simulations corresponding to the different panels of Figure 1.

1.1 Sampling effort

We start by drawing a Directed Acyclic Graph (DAG). Note that we represent unobserved variables that were not shown, or were left implicit, in the manuscript’s figure.

DAG illustrating the assumed causal structure of Figure 1A.
Here, \(y_{[a,b]}\) represents the observed number of interactions from \(a\) to \(b\);
\(m_{[a,b]}\) is the true rate of interactions from \(a\) to \(b\);
\(S_{[a]}\) and \(S_{[b]}\) represent individual-level sampling efforts;
\(S_{[a,b]}\) is the dyad-level sampling effort;
\(\gamma_{[a]}\) is an unobserved variable affecting the tendency of an individual, \(a\) to give more or fewer interactions across partners;
\(\rho_{[a]}\) is an unobserved variable affecting the tendency of an individual, \(a\), to receive more or fewer interactions across partners;
and \(\tau\) is an unobserved variable affecting the tendency of an individual, \(a\), to give more or fewer interactions to a specific partner, \(b\).

We then write a generative model corresponding to the DAG:

\[ \begin{align} f_{S_{[a]}} : \hspace{0.5cm} & S_{[a]} \sim \mathrm{Uniform}(0, 10)\\ f_{S_{[a, b]}} : \hspace{0.5cm} & S_{[a, b]} = S_{[a]} + S_{[b]}\\ \\ f_{\gamma} : \hspace{0.5cm} & \gamma_{[a]} \sim \mathrm{Normal}(0, \sigma_{\gamma})\\ f_{\rho} : \hspace{0.5cm} & \rho_{[a]} \sim \mathrm{Normal}(0, \sigma_{\rho})\\ f_{\tau} : \hspace{0.5cm} & \tau_{[a, b]} \sim \mathrm{Normal}(0, \sigma_{\tau}) \\ \\ f_{m} : \hspace{0.5cm} & m_{[a, b]} = \mathrm{exp}(\delta + \gamma_{[a]} + \rho_{[b]} + \tau_{[a, b]})\\ f_{y} : \hspace{0.5cm} & y_{[a, b]} \sim \mathrm{Poisson}(m_{[a, b]} \cdot S_{[a, b]})\\ \end{align} \]

scm.1 <- function(
  # Observed features
  N_ind = 20, # Nb of individuals
  N_dyad = ((N_ind * N_ind) - N_ind) / 2, # Nb of dyads

  # Structural paramaters
  delta = 0.2, # Baseline interaction rate (intercept)
  sigma_gamma = 0.1, # SD of gamma
  sigma_rho = 0.1, # SD of rho
  sigma_tau = 0.1 # SD of tau
  ){
  
    ## Generate data
    ID_features <- tibble(
      ID = c(1:N_ind),
      gamma = rnorm(N_ind, 0, sigma_gamma),
      rho = rnorm(N_ind, 0, sigma_rho),
      S = runif(N_ind, 0, 10)
    )
    
    # Assign each directed dyad...
    dyad_features <- tibble(
      ind_a = t(combn(N_ind, 2))[, 1],
      ind_b = t(combn(N_ind, 2))[, 2],
      
      # a dyad ID and tau for each direction
      dyad = c(1:N_dyad),
      tau_ab = rnorm(N_dyad, 0, sigma_tau),
      tau_ba = rnorm(N_dyad, 0, sigma_tau)
    )
    
    # Combine individual and dyadic features
    df <- dyad_features %>%
      # Add A features
      left_join(ID_features, by = c("ind_a" = "ID")) %>%
      rename(gamma_a = gamma, rho_a = rho, S_a = S) %>%
      
      # Add B features
      left_join(ID_features, by = c("ind_b" = "ID")) %>%
      rename(gamma_b = gamma, rho_b = rho, S_b = S) %>%
      
      # Dyadic features, compute rate
      mutate(S_ab = S_a + S_b,
             m_ab = exp(delta + gamma_a + rho_b + tau_ab),
             m_ba = exp(delta + gamma_b + rho_a + tau_ba)) %>%
      
      # Generate observations
      mutate(y_ab = rpois(n(), m_ab * S_ab),
             y_ba = rpois(n(), m_ba * S_ab))
    
    ## Output
    df %>%
      return()
  }

We run the function, and plot \(y_{[a, b]}\) against \(y_{[b, a]}\).

df1 <- scm.1()
df1 %>%
  ggplot(aes(y_ab, y_ba)) +
  geom_point(
    fill = '#628b96',
    shape = 21,
    color = "#f0f2f2",
    size = 4,
    alpha = 0.8
  ) +
  labs(x = expression(y["ab"]),
       y = expression(y["ba"]))

In the following subsections, we follow the same workflow (DAG \(\to\) generative model \(\to\) plot synthetic data) for different causal structures.

1.2 Individual-level causal effects

Directed Acyclic Graph:

DAG illustrating the assumed causal structure of Figure 1B. \(X_{[a]}\) and \(X_{[b]}\) are individual-level phenotypic features (e.g., age, colouration).

Associated generative model:

\[ \begin{align} f_{X} : \hspace{0.5cm} & X_{[a]} \sim \mathrm{Normal}(0, \sigma_{X})\\ \\ f_{\gamma} : \hspace{0.5cm} & \gamma_{[a]} \sim \mathrm{Normal}(\beta_{\gamma} \cdot X_{[a]}, \sigma_{\gamma})\\ f_{\rho} : \hspace{0.5cm} & \rho_{[a]} \sim \mathrm{Normal}(\beta_{\rho} \cdot X_{[a]}, \sigma_{\rho})\\ f_{\tau} : \hspace{0.5cm} & \tau_{[a, b]} \sim \mathrm{Normal}(0, \sigma_{\tau}) \\ \\ f_{m} : \hspace{0.5cm} & m_{[a, b]} = \mathrm{exp}(\delta + \gamma_{[a]} + \rho_{[b]} + \tau_{[a, b]})\\ f_{y} : \hspace{0.5cm} & y_{[a, b]} \sim \mathrm{Poisson}(m_{[a, b]})\\ \end{align} \]

scm.2 <- function(
  # Observed features
  N_ind = 20, # Nb of individuals
  N_dyad = ((N_ind * N_ind) - N_ind) / 2, # Nb of dyads

  # Structural paramaters
  delta = 1, # Baseline interaction rate (intercept)
  beta_g = 0.7, # causal effect gamma
  beta_r = 0.7, # causal effect rho
  sigma_gamma = 0.2, # SD of gamma
  sigma_rho = 0.2, # SD of rho
  sigma_tau = 0.2 # SD of tau
  ){
  
    ## Generate data
    ID_features <- tibble(
      ID = c(1:N_ind),
      X = rnorm(N_ind),
      gamma = rnorm(N_ind, beta_g * X, sigma_gamma),
      rho = rnorm(N_ind, beta_r * X, sigma_rho)
    )
    
    # Assign each directed dyad...
    dyad_features <- tibble(
      ind_a = t(combn(N_ind, 2))[, 1],
      ind_b = t(combn(N_ind, 2))[, 2],
      
      # a dyad ID and tau for each direction
      dyad = c(1:N_dyad),
      tau_ab = rnorm(N_dyad, 0, sigma_tau),
      tau_ba = rnorm(N_dyad, 0, sigma_tau)
    )
    
    # Combine individual and dyadic features
    df <- dyad_features %>%
      # Add A features
      left_join(ID_features, by = c("ind_a" = "ID")) %>%
      rename(gamma_a = gamma, rho_a = rho) %>%
      
      # Add B features
      left_join(ID_features, by = c("ind_b" = "ID")) %>%
      rename(gamma_b = gamma, rho_b = rho) %>%
      
      # Rate
      mutate(m_ab = exp(delta + gamma_a + rho_b + tau_ab),
             m_ba = exp(delta + gamma_b + rho_a + tau_ba)) %>%
      
      # Generate observations
      mutate(y_ab = rpois(n(), m_ab),
             y_ba = rpois(n(), m_ba))
    
    ## Output
    df %>%
      return()
  }

We run the function, and plot \(y_{[a, b]}\) against \(y_{[b, a]}\).

set.seed(666)
df2 <- scm.2()
df2 %>%
  ggplot(aes(y_ab, y_ba))  +
  geom_point(
    fill = '#628b96',
    shape = 21,
    color = "#f0f2f2",
    size = 4,
    alpha = 0.8
  ) +
  labs(x = expression(y["ab"]),
       y = expression(y["ba"]))

We show the number of observed interactions \(y_{[a, b]}\) for five recipients \(b\) (x-axis) and five actors \(a\) (one colour per actor \(a\)).

# We run the function
  set.seed(666)
  df2.2 <- scm.2(
      N_ind = 5,
      sigma_gamma = 0.5,
      beta_r = 0.2
    )

# Reorganise data
  df2.2 %>% 
    select(ind_a, ind_b, y_ab) %>%
    rename(actor = ind_a,
           Recipient = ind_b,
           y = y_ab) %>%
    bind_rows(
      df2.2 %>% 
        select(ind_a, ind_b, y_ba) %>%
        rename(actor = ind_b,
               Recipient = ind_a,
               y = y_ba)) %>%
    # Plot
    ggplot(aes(x = Recipient, y = y,
      group = as.factor(actor))) +
    geom_line(aes(color = as.factor(actor)), linewidth = 1.25) +
    geom_point(aes(fill = as.factor(actor)),
      color = "#f0f2f2",
      size = 4,
      shape = 21,
      stroke = 1) +
    theme(
      legend.position = "none",
      panel.grid.major.x = element_line(color = "#ebe9df", linewidth = 0.3)) +
    scale_color_manual(
      values = c("#BF402B", "#FECD64", "#C5DAEF", "#2071B3", "#389E55")) +
    scale_fill_manual(
      values = c("#BF402B", "#FECD64", "#C5DAEF", "#2071B3", "#389E55"))

Then, the opposite: we show the number of observed interactions \(y_{[a, b]}\) for five actors \(a\) (x-axis) and five recipients \(b\) (one colour per recipient \(b\)).

# We run the function
  set.seed(666)
  df2.3 <- scm.2(
      N_ind = 5,
      sigma_rho = 0.5,
      beta_g = 0.2
    )
  
# Reorganise data
  df2.3 %>% 
    select(ind_a, ind_b, y_ab) %>%
    rename(Actor = ind_a,
           recipient = ind_b,
           y = y_ab) %>%
    bind_rows(
      df2.3 %>% 
        select(ind_a, ind_b, y_ba) %>%
        rename(Actor = ind_b,
               recipient = ind_a,
               y = y_ba)) %>%
    
    # Plot
    ggplot(aes(x = Actor, y = y, group = as.factor(recipient))) +
    geom_line(aes(color = as.factor(recipient)), linewidth = 1.25) +
    geom_point(aes(fill = as.factor(recipient)),
      color = "#f0f2f2",
      size = 4,
      shape = 21,
      stroke = 1) +
    theme(legend.position = "none",
      panel.grid.major.x = element_line(color = "#ebe9df", linewidth = 0.3)) +
    scale_color_manual(
      values = c("#BF402B", "#FECD64", "#C5DAEF", "#2071B3", "#389E55")) +
    scale_fill_manual(
      values = c("#BF402B", "#FECD64", "#C5DAEF", "#2071B3", "#389E55"))

1.3 Dyad-level causal effects

Directed Acyclic Graph:

DAG illustrating the assumed causal structure of Figure 1C. \(R_{[a]}\) and \(R_{[b]}\) are the individual-level phenotypic features (e.g., dominance rank), and \(\Delta R_{|a, b|}\) is a dyad-level feature (e.g., difference in rank).

Associated generative model:

\[ \begin{align} f_{R} : \hspace{0.5cm} & R_{[a]} \sim \mathrm{DUniform}(0, N)\\ \\ f_{\gamma} : \hspace{0.5cm} & \gamma_{[a]} \sim \mathrm{Normal}(0, \sigma_{\gamma})\\ f_{\rho} : \hspace{0.5cm} & \rho_{[a]} \sim \mathrm{Normal}(0, \sigma_{\rho})\\ f_{\tau} : \hspace{0.5cm} & \tau_{[a, b]} \sim \mathrm{Normal}(\beta_{R} \cdot (R_{[b]} - R_{[a]}), \sigma_{\tau}) \\ \\ f_{m} : \hspace{0.5cm} & m_{[a, b]} = \mathrm{exp}(\delta + \gamma_{[a]} + \rho_{[b]} + \tau_{[a, b]})\\ f_{y} : \hspace{0.5cm} & y_{[a, b]} \sim \mathrm{Poisson}(m_{[a, b]})\\ \end{align} \]

scm.3 <- function(
  # Observed features
  N_ind = 20, # Nb of individuals
  N_dyad = ((N_ind * N_ind) - N_ind) / 2, # Nb of dyads

  # Structural paramaters
  delta = 2, # Baseline interaction rate (intercept)
  beta = 1, # Effect of rank
  sigma_gamma = 0.2, # SD of gamma
  sigma_rho = 0.2, # SD of rho
  sigma_tau = 0.2 # SD of tau
  ){
  
    ## Generate data
    ID_features <- tibble(
      ID = c(1:N_ind),
      R = (sample(c(1:N_ind), N_ind, replace = FALSE)) / N_ind,
      gamma = rnorm(N_ind, 0, sigma_gamma),
      rho = rnorm(N_ind, 0, sigma_rho)
    )
    
    # Assign each directed dyad...
    dyad_features <- tibble(
      ind_a = t(combn(N_ind, 2))[, 1],
      ind_b = t(combn(N_ind, 2))[, 2],
      
      # a dyad ID and tau for each direction
      dyad = c(1:N_dyad),
    )
    
    # Combine individual and dyadic features
    df <- dyad_features %>%
      # Add A features
      left_join(ID_features, by = c("ind_a" = "ID")) %>%
      rename(gamma_a = gamma, rho_a = rho, Ra = R) %>%
      
      # Add B featues
      left_join(ID_features, by = c("ind_b" = "ID")) %>%
      rename(gamma_b = gamma, rho_b = rho, Rb = R) %>%
      
      # dyadic features and compute rates
      mutate(tau_ab = rnorm(N_dyad, beta * (Rb - Ra), sigma_tau),
             tau_ba = rnorm(N_dyad, beta * (Ra - Rb), sigma_tau),
             m_ab = exp(delta + gamma_a + rho_b + tau_ab),
             m_ba = exp(delta + gamma_b + rho_a + tau_ba)) %>%
      
      # Generate observations
      mutate(y_ab = rpois(n(), m_ab),
             y_ba = rpois(n(), m_ba))
    
    ## Output
    df %>%
      return()
  }

We run the function, and plot \(y_{[a, b]}\) against \(y_{[b, a]}\).

set.seed(30)
df3 <- scm.3()
df3 %>%
  ggplot(aes(y_ab, y_ba)) +
  geom_point(
    fill = '#628b96',
    shape = 21,
    color = "#f0f2f2",
    size = 4,
    alpha = 0.8
  ) +
  labs(x = expression(y["ab"]),
       y = expression(y["ba"]))

1.4 Dyadic reciprocity

Directed Acyclic Graph:

DAG illustrating the assumed causal structure of Figure 1D. \(y_{[a,b]}(t)\) represents the number of observed interactions from \(a\) to \(b\) at time \(t\), and \(y_{[a,b]}(t+1)\) the number of observed interactions from \(a\) to \(b\) at time \(t+1\).

Associated generative model:

\[ \begin{align} \text{For } t = 1:\\ m_{[a, b, t]} &= 1\\ m_{[b, a, t]} &= 1\\ y_{[a, b, t]} &\sim \mathrm{Poisson}(m_{[a, b, t]})\\ y_{[b, a, t]} &\sim \mathrm{Poisson}(m_{[b, a, t]})\\ \\ \text{For } t = 2, \dots, T:\\ m_{[a, b, t]} &= m_{[a, b, t-1]} + \epsilon_{[a, b, t]} + \begin{cases} +1 & \text{if } y_{[b, a, t-1]} > 4 \text{ and } m_{[a, b, t]} < K\\ -1 & \text{otherwise.} \end{cases}\\ y_{[a, b, t]} &\sim \begin{cases} \mathrm{Poisson}(m_{[a, b, t]}) & \text{if } m_{[a, b, t]} > 1\\ \mathrm{Poisson}(1) & \text{if } m_{[a, b, t]} \leq 1 \end{cases}\\ \\ m_{[b, a, t]} &= m_{[b, a, t-1]} + \epsilon_{[b, a, t]} + \begin{cases} +1 & \text{if } y_{[a, b, t-1]} > 4 \text{ and } m_{[b, a, t]} < K\\ -1 & \text{otherwise.} \end{cases}\\ y_{[b, a, t]} &\sim \begin{cases} \mathrm{Poisson}(m_{[b, a, t]}) & \text{if } m_{[b, a, t]} > 1\\ \mathrm{Poisson}(1) & \text{if } m_{[b, a, t]} \leq 1 \end{cases}\\ \epsilon_{[a, b, t]}, \epsilon_{[b, a, t]} &\sim \mathrm{Normal}(0, 1.5) \end{align} \]

scm.4 <- function(
  N_ind = 20, # Nb of individuals
  N_dyad = ((N_ind * N_ind) - N_ind) / 2, # Nb of dyads
  delta = 1, # Minimal and initial interaction rate
  nb_loops = 500
){

  # Empty data frame with name and dyad id
  basic_df <- tibble(
    ind_a = t(combn(N_ind, 2))[, 1],
    ind_b = t(combn(N_ind, 2))[, 2],
    dyad = c(1:N_dyad)
  )
  
  # Data frame at step t = 1
  list <- list()
  list[[1]] <- basic_df %>% mutate(
    m_ab = rpois(N_dyad, delta),
    m_ba = rpois(N_dyad, delta),
    y_ab = rpois(N_dyad, m_ab),
    y_ba = rpois(N_dyad, m_ba),
    y_ab_z = (y_ab - mean(c(y_ab, y_ba))) / sd(c(y_ab, y_ba)),
    y_ba_z = (y_ba - mean(c(y_ab, y_ba))) / sd(c(y_ab, y_ba)),
    t = 1
  )
  
  # Data frame at steps t > 1
  for (i in 2:nb_loops) {
    list[[i]] <- basic_df %>% mutate(
      m_ab = list[[i - 1]]$m_ab +
             ifelse(list[[i - 1]]$y_ba > 4 & list[[i - 1]]$m_ab < 10, 1, -1) +
             rnorm(N_dyad, 0, 1.5),
      m_ab = ifelse(m_ab < delta, delta, m_ab),
      m_ba = list[[i - 1]]$m_ba +
             ifelse(list[[i - 1]]$y_ab > 4 &
             list[[i - 1]]$m_ba < 10, 1, -1) +
             rnorm(N_dyad, 0, 1.5),
      m_ba = ifelse(m_ba < delta, delta, m_ba),
      y_ab = rpois(N_dyad, m_ab),
      y_ba = rpois(N_dyad, m_ba),
      y_ab_z = (y_ab - median(c(y_ab, y_ba))) / sd(c(y_ab, y_ba)),
      y_ba_z = (y_ba - median(c(y_ab, y_ba))) / sd(c(y_ab, y_ba)),
      t = i
    )
  } # i
 list %>% bind_rows() %>% return() 
} # fct

We run the function, and show how the number of interactions in one direction \(y_{[a, b]}\) (blue) and the opposite direction \(y_{[b, a]}\) (yellow), on the y-axis, change as a function of time (x-axis) for 16 dyads (one per panel).

# Run the function
  set.seed(2666)
  df4 <- scm.4(nb_loops = 200)

# Plot
  df4 %>%
  filter(dyad < 17) %>%
      gather(direction, y, m_ab:m_ba) %>%
      ggplot(aes(x = t, y = y, group = direction, color = as.factor(direction))) +
      geom_line(alpha = 0.7) +
      facet_wrap(~ dyad) +
      scale_color_manual(values = c("#d3c94e", "#3d6a77")) +
      theme(legend.position = "none",
            strip.background = element_blank(), 
            strip.text = element_blank()) +
    labs(x = "Time", y = "")

We also plot \(y_{[a, b]}\) against \(y_{[b, a]}\) in a cross-sectional sample.

df4 %>%
  filter(t == 200) %>%
  ggplot(aes(y_ab, y_ba)) +
  geom_point(
    fill = '#628b96',
    shape = 21,
    color = "#f0f2f2",
    size = 4,
    alpha = 0.8
  ) +
  labs(x = expression(y["ab"]),
       y = expression(y["ba"]))

1.5 Triadic closure

Below, we represent the causal process with a DAG.

DAG illustrating the assumed causal structure of Figure 1E. For all triads \((a, b, c)\) in the population understudy, \(y_{|a, b|}\) represents the number of (undirected) observed interactions between \(a\) and \(b\), \(y_{|a, c|}\) the number of observed interactions between \(a\) and \(c\), and \(y_{|b, c|}\) the number of observed interactions between \(b\) and \(c\). Like in the previous subsection, these variables are indexed by time \(t\).

Note that compared to the causal diagram in the manuscript, we show the process for all individuals in the triads (\(y_{|ab|} \rightarrow y_{|bc|} \leftarrow y_{|ac|}\), \(y_{|ac|} \rightarrow y_{|ab|} \leftarrow y_{|bc|}\), and \(y_{|ab|} \rightarrow y_{|ac|} \leftarrow y_{|bc|}\)). We do not explicitly represent the intermediate variables \(k\) for clarity.

Associated generative model:

\[ \begin{align} \text{For } t = 1\\ \text{Step one:} & \hspace{1cm} y_{|a, b, t, 1|} \sim \mathrm{Bernoulli}(p_{\mathrm{initial}})\\ \text{Step two:} & \hspace{1cm} y_{|a, b, t, 2|} = y_{|a, b, t, 1|}\\ \\ \text{For } t = 2, \dots, T\\ \text{Step one:} & \hspace{1cm} y_{|a, b, t, 1|} \sim \begin{cases} \text{Bernoulli}(p_{0 \rightarrow 1}), &\text{ if } y_{|a, b, t - 1, 2|} = 0\\ \text{Bernoulli}(p_{1 \rightarrow 0}), &\text{ if } y_{|a, b, t - 1, 2|} = 1\\ \end{cases}\\ \text{Step two:} & \hspace{1cm} \text{\textbf{For }} N \text{ edges } y_{|a, b, t, 1|} = 1 \text{ selected at random}, \textbf{do}\\& \hspace{1.5cm} \text{Select an individual } c \text{ at random},\\ & \hspace{2cm} \text{\textbf{If} } y_{|a, b, t, 1|} + y_{|a, c, t, 1|} + y_{|b, c, t, 1|} = 2 \text{, \textbf{then}}\\ & \hspace{2.5cm} \begin{cases} y_{|a, b, t, 2|} = 1\\ y_{|a, c, t, 2|} = 1\\ y_{|b, c, t, 2|} = 1 \end{cases}\\ & \hspace{2cm} \textbf{else}\\ & \hspace{2.5cm} \begin{cases} y_{|a, b, t, 2|} = y_{|a, b, t, 1|}\\ y_{|a, c, t, 2|} = y_{|a, c, t, 1|}\\ y_{|b, c, t, 2|} = y_{|b, c, t, 1|} \end{cases}\\ & \hspace{2cm} \textbf{end if}\\ & \hspace{1.5cm} \textbf{end for} \end{align} \]

scm.5 <- function(
    N_ind = 12, # Nb of individuals
    N_dyad = ((N_ind * N_ind) - N_ind) / 2, # Nb of dyads
    N_triad = t(combn(N_ind, 3))[, 1] %>% length(), #Nb triads
    p_initial = 0.2, # Inital proba of forming a tie y
    p_sw_1_0 = (1/200), # switch from 1 to 0 (loss of tie)
    p_sw_0_1 = (1/2000), # switch from 0 to 1 (new random tie)
    n_l_i = 100, # Number of iterations i (time)
    n_l_j = 15){ # Number of considered triads undergoing triadic closure 
    
### Create objects
# Empty data frame with name and dyad id
  basic_df <- tibble(
    ind_a = t(combn(N_ind, 2))[, 1],
    ind_b = t(combn(N_ind, 2))[, 2],
    dyad = paste0(ind_a, "_", ind_b)
  )
  
  list <- list() # list containing data frames at different t
  list[[1]] <- list() # list containing data frames at different t
  list[[2]] <- list() # list containing data frames at different t
  d <- list

### Iteration 1
## Step 1
# Edge generated with fixed probability
  (list[[1]][[1]] <- basic_df %>% mutate(
    y_ab = rbinom(N_dyad, 1, p_initial),
    t = 1
  ))
    
## STEP 2
# No triangle closure during the first iteration
  list[[2]][[1]] <- list[[1]][[1]]
    
### Iterations i
  for (i in 2:n_l_i){
    ## STEP 1
    # Random creation and losses of tie
    list[[1]][[i]] <- list[[2]][[i-1]] %>% mutate(
      rs = runif(n(), 0, 1),
      y_ab = case_when(
        rs < p_sw_1_0 & y_ab == 1 ~ 0,
        rs >= p_sw_1_0 & y_ab == 1 ~ 1,
        rs < p_sw_0_1 & y_ab == 0 ~ 1,
        rs >= p_sw_0_1 & y_ab == 0 ~ 0),
      t = i
    ) %>%
      select(-rs)
    
    ## STEP 2
    # Note that there is a non-null prob. that the same triad
    # is selected twice in the same iteration, in which case
    # nothing happens: the triad is already closed
    for (j in 1:n_l_j){
      # Randomly select one dyad a-b
      (dyad_ab <- list[[1]][[i]] %>%
         filter(y_ab == 1) %>%
         slice_sample(n = 1))
      
      # Save the a and b's ids
      (dyad_ab_ids <- dyad_ab %>%
          select(ind_a, ind_b) %>%
          unlist())
      
      # Sample one individual c among all their potential partners
      (id_c <- sample(c(1:N_ind)[-c(dyad_ab_ids)], 1))
      
      # Select dyad a-c
      (dyad_ac <- list[[1]][[i]] %>%
          filter(dyad == paste0(id_c, "_", dyad_ab_ids[1]) |
                   dyad == paste0(dyad_ab_ids[1], "_", id_c)))
      
      # Select dyad b-c
      (dyad_bc <- list[[1]][[i]] %>%
          filter(dyad == paste0(id_c, "_", dyad_ab_ids[2]) |
                   dyad == paste0(dyad_ab_ids[2], "_", id_c)))
      
      ## Data frame at iteration i, step 2:
      # If two out of the three edges exist, close the triangle
      if (dyad_ab$y_ab + dyad_ac$y_ab + dyad_bc$y_ab == 2) {
        list[[2]][[i]] <- list[[1]][[i]] %>%
          mutate(y_ab = ifelse(dyad == dyad_ab$dyad |
                                 dyad == dyad_ac$dyad |
                                 dyad == dyad_bc$dyad, 
                               1, y_ab))
        # Otherwise, no change
      } else {list[[2]][[i]] <- list[[1]][[i]]}
    } # j
  
  # Reformat data
  (d[[i]] <- list[[2]][[i]] %>%
      select(- c(t, dyad)) %>%
      rename(y = y_ab, ind_i = ind_a, ind_j = ind_b))
  
  
  # Assign each triad...
  d[[i]] <- tibble(
    ind_a = t(combn(N_ind, 3))[, 1],
    ind_b = t(combn(N_ind, 3))[, 2],
    ind_c = t(combn(N_ind, 3))[, 3],
    
    # a tried ID
    triad = c(1:N_triad)
  ) %>%
    left_join(d[[i]], by = c("ind_a" = "ind_i", "ind_b" = "ind_j")) %>%
    rename(y_ab = y) %>%
    
    left_join(d[[i]], by = c("ind_b" = "ind_i", "ind_c" = "ind_j")) %>%
    rename(y_bc = y) %>%
    
    left_join(d[[i]], by = c("ind_a" = "ind_i", "ind_c" = "ind_j")) %>%
    rename(y_ac = y)
  } 
  
  output <- list(
    list = list,
    d = d
  )
  
  return(output)
} # scm.5

We run the function:

set.seed(123)
df5 <- scm.5(N_ind = 12,
             p_sw_1_0 = (1/180))

Plot social network:

Show code:

library(animation)
saveGIF({
  for (i in 1:100) {
    # Filter data to include only ties with y_ab == 1
    ties <- subset(df5$list[[2]][[i]], y_ab == 1)
    # Create a graph object
    graph <- tbl_graph(edges = ties, directed = FALSE)
    p <- ggraph(graph, 'linear', circular = TRUE) +
      geom_edge_link(color = "#3A5B71") +
      geom_node_point(
        fill = '#3D485B',
        shape = 21,
        color = "#f0f2f2",
        size = 5
      ) +
      labs(title = paste0("Time step ", i)) +
      theme_void() 
    print(p)
  }
}, interval = .2, movie.name = ".gif",
ani.width = 1500, ani.height = 1500, ani.res = 600)

We then look at the association created by the causal process. Specifically, we plot the association between \(y_{|a,b|}(t) + y_{|a,c|}(t)\) and \(y_{|b,c|}(t)\).

df5$d[[100]]$past_sum <- df5$d[[100]]$y_ab * df5$d[[100]]$y_ac

df5$d[[100]] %>%
  ggplot(aes(x = past_sum, y = y_bc)) +
  geom_jitter(width = 0.1, height = 0.1, size = 5,
              fill = "#628b96",
               pch = 21,
               colour = "white",
               alpha = 0.6) +
    scale_x_continuous(breaks = c(0, 1), limits = c(-0.2, 1.2)) +
  scale_y_continuous(breaks = c(0, 1), limits = c(-0.2, 1.2)) +
  labs(x = "y_ab(t) × y_ac(t)",
       y = "y_ac(t)")

2 Uneven sampling

In this section, we present a generative model, generate synthetic data from it, and compare the estimates from two different statistical models. In doing so, we show that failing to block the backdoor path on the DAG leads to a biased estimate for the causal effect of interest.

Directed Acyclic Graph:

DAG illustrating the assumed causal structure of Figure 2. \(X_{[a]}\) and \(X_{[b]}\) are individual-level phenotypic features (e.g., colouration); \(S_{[a]}\) and \(S_{[b]}\) represent individual-level sampling effort;
\(S_{[a,b]}\) is the dyad-level sampling effort.

Associated Generative Model:

\[ \begin{align} f_{X} : \hspace{0.5cm} & X_{[a]} \sim \mathrm{Normal}(0, \sigma_{X})\\ f_{S_{[a]}} : \hspace{0.5cm} & S_{[a]} \sim \mathrm{LogNormal}(0.2 \cdot \exp (X_{[a]}), \sigma_{S})\\ f_{S_{[a, b]}} : \hspace{0.5cm} & S_{[a, b]} = S_{[a]} + S_{[b]}\\ \\ f_{\gamma} : \hspace{0.5cm} & \gamma_{[a]} \sim \mathrm{Normal}(\beta_{\gamma} \cdot X_{[a]}, \sigma_{\gamma})\\ f_{\rho} : \hspace{0.5cm} & \rho_{[a]} \sim \mathrm{Normal}(0, \sigma_{\rho})\\ f_{\tau} : \hspace{0.5cm} & \tau_{[a, b]} \sim \mathrm{Normal}(0, \sigma_{\tau}) \\ \\ f_{m} : \hspace{0.5cm} & m_{[a, b]} = \mathrm{exp}(\delta + \gamma_{[a]} + \rho_{[b]} + \tau_{[a, b]})\\ f_{y} : \hspace{0.5cm} & y_{[a, b]} \sim \mathrm{Poisson}(m_{[a, b]} \cdot S_{[a, b]})\\ \end{align} \]

Code for the data simulation:

scm.6 <- function(
  # Observed features
  N_ind = 20, # Nb of individuals
  N_dyad = ((N_ind * N_ind) - N_ind) / 2, # Nb of dyads

  # Structural parameters
  delta = 0.2, # Baseline interaction rate (intercept)
  beta_g = 0.5, # causal effect gamma
  sigma_gamma = 0.1, # SD of gamma
  sigma_rho = 0.1, # SD of rho
  sigma_tau = 0.1 # SD of tau
  ){
  
    ## Generate data
    ID_features <- tibble(
      ID = c(1:N_ind),
      x = rnorm(N_ind),
      gamma = rnorm(N_ind, beta_g * x, sigma_gamma),
      rho = rnorm(N_ind, 0, sigma_rho),
      S = rlnorm(N_ind, 0.2 * exp(x), 0.2),
    )
    
    # Assign each directed dyad...
    dyad_features <- tibble(
      ind_a = t(combn(N_ind, 2))[, 1],
      ind_b = t(combn(N_ind, 2))[, 2],
      
      # a dyad ID and tau for each direction
      dyad = c(1:N_dyad),
      tau_ab = rnorm(N_dyad, 0, sigma_tau),
      tau_ba = rnorm(N_dyad, 0, sigma_tau)
    )
    
    # Combine individual and dyadic features
    df <- dyad_features %>%
      # Add A features
      left_join(ID_features, by = c("ind_a" = "ID")) %>%
      rename(gamma_a = gamma, rho_a = rho, S_a = S,
             x_a = x) %>%
      
      # Add B featues
      left_join(ID_features, by = c("ind_b" = "ID")) %>%
      rename(gamma_b = gamma, rho_b = rho, S_b = S,
             x_b = x) %>%
      
      # Rate
      mutate(S_ab = S_a + S_b,
             m_ab = exp(delta + gamma_a + rho_b + tau_ab),
             m_ba = exp(delta + gamma_b + rho_a + tau_ba)) %>%
      
      # Generate observations
      mutate(y_ab = rpois(n(), m_ab * S_ab),
             y_ba = rpois(n(), m_ba * S_ab))
    
    ## Output
    df %>%
      return()
  }

We run the function above:

set.seed(3666)
df6 <- scm.6()
d6 <- df6 %>% as.list()
d6$N <- nrow(df6)
d6$N_ind <- 20
d6$N_dyad <- 190

Statistical models:

# Stan model
(m_6_adj <- cmdstan_model("./stan_models/adjusted_by_S.stan"))

data{
  // Indices
  int<lower = 1> N; // Nb of observations
  int<lower = 1> N_ind; // Nb of individuals
  int<lower = 1> N_dyad; // Nb of non-directed dyads
  
  // Observables
  array[N] int ind_a; // Individual a
  array[N] int ind_b; // Individual b
  array[N] int dyad; // non-directed dyad
  array[N] int y_ab; // Interactions from a to b
  array[N] int y_ba; // Interactions from b to a
  vector[N] x_a; // Value of x for actor
  vector[N] x_b; // Value of x for receiver
  vector[N] S_ab; // Sampling effort for the dyad
}

parameters{
  // Fixed effects
  real D;
  real b_G;

  // Individual varying effects
  matrix[2, N_ind] z_K; // z_G and z_R (horizontal)
  real<lower = 0> s_G; // SD of G
  real<lower = 0> s_R; // SD of R
  cholesky_factor_corr[2] L_ind; // Cholesky factor of corr. matrix
  
  // Dyadic varying effects
  matrix[2, N_dyad] z_T; // z_T_ab and z_T_ba (wide format)
  real<lower = 0> s_T; // Unique SD of directed ties
  cholesky_factor_corr[2] L_dyad; // Cholesky factor of corr. matrix
}

transformed parameters{
  // Individual varying effects 
    matrix[N_ind, 2] K; // K matrix (i.e. G and R): vertical format
    K = (diag_pre_multiply([s_G, s_R], L_ind) * z_K)';
    
  // Rename individual random effects
  vector[N_ind] g = K[, 1]; // individual giving random effect
  vector[N_ind] r = K[, 2]; // individual receiving random effect
  
  // Compute gen. giving and receiving
  vector[N_ind] G;
  vector[N_ind] R;
  for (j in 1:N){
    G[ind_a[j]] = g[ind_a[j]] + b_G * x_a[j]; // Generalised giving (a)
    G[ind_b[j]] = g[ind_b[j]] + b_G * x_b[j]; // Generalised giving (b)
    
    R[ind_a[j]] = r[ind_a[j]]; // Generalised receiving (a)
    R[ind_b[j]] = r[ind_b[j]]; // Generalised receiving (b)
  }
    
  // Dyadic varying effects 
    matrix[N_dyad, 2] T; // T_ab and T_ba (vertical format)
    T = (diag_pre_multiply(rep_vector(s_T, 2), L_dyad) * z_T)';
    
    // Rename dyadic random effects
    vector[N_dyad] T_ab = T[, 1]; // T_ab
    vector[N_dyad] T_ba = T[, 2]; // T_ba
  
  
  // Expected dyadic rates
  vector[N] m_ab;
  vector[N] m_ba;
  
  // Two parallel linear models
  m_ab = exp(D + G[ind_a] + R[ind_b] + T_ab[dyad]);
  m_ba = exp(D + G[ind_b] + R[ind_a] + T_ba[dyad]);
}

model{
  // Priors
    // Fixed effects
    D ~ normal(0, 1);
    b_G ~ normal(0, 1);
    
    // Varying individual effects
    to_vector(z_K) ~ normal(0, 1);
    s_G ~ exponential(1);
    s_R ~ exponential(1);
    L_ind ~ lkj_corr_cholesky(2);
    
    // Varying dyadic effects
    to_vector(z_T) ~ normal(0, 1);
    s_T ~ exponential(1);
    L_dyad ~ lkj_corr_cholesky(2);
  
  // Likelihood
  y_ab ~ poisson(m_ab .* S_ab);
  y_ba ~ poisson(m_ba .* S_ab);
}

(m_6_unadj <- cmdstan_model("./stan_models/not_adjusted_by_S.stan"))

data{
  // Indices
  int<lower = 1> N; // Nb of observations
  int<lower = 1> N_ind; // Nb of individuals
  int<lower = 1> N_dyad; // Nb of non-directed dyads
  
  // Observables
  array[N] int ind_a; // Individual a
  array[N] int ind_b; // Individual b
  array[N] int dyad; // non-directed dyad
  array[N] int y_ab; // Interactions from a to b
  array[N] int y_ba; // Interactions from b to a
  vector[N] x_a; // Value of x for actor
  vector[N] x_b; // Value of x for receiver
}

parameters{
  // Fixed effects
  real D;
  real b_G;

  // Individual varying effects
  matrix[2, N_ind] z_K; // z_G and z_R (horizontal)
  real<lower = 0> s_G; // SD of G
  real<lower = 0> s_R; // SD of R
  cholesky_factor_corr[2] L_ind; // Cholesky factor of corr. matrix
  
  // Dyadic varying effects
  matrix[2, N_dyad] z_T; // z_T_ab and z_T_ba (wide format)
  real<lower = 0> s_T; // Unique SD of directed ties
  cholesky_factor_corr[2] L_dyad; // Cholesky factor of corr. matrix
}

transformed parameters{
  // Individual varying effects 
    matrix[N_ind, 2] K; // K matrix (i.e. G and R): vertical format
    K = (diag_pre_multiply([s_G, s_R], L_ind) * z_K)';
    
  // Rename individual random effects
  vector[N_ind] g = K[, 1]; // individual giving random effect
  vector[N_ind] r = K[, 2]; // individual receiving random effect
  
  // Compute gen. giving and receiving
  vector[N_ind] G;
  vector[N_ind] R;
  for (j in 1:N){
    G[ind_a[j]] = g[ind_a[j]] + b_G * x_a[j]; // Generalised giving (a)
    G[ind_b[j]] = g[ind_b[j]] + b_G * x_b[j]; // Generalised giving (b)
    
    R[ind_a[j]] = r[ind_a[j]]; // Generalised receiving (a)
    R[ind_b[j]] = r[ind_b[j]]; // Generalised receiving (b)
  }
    
  // Dyadic varying effects 
    matrix[N_dyad, 2] T; // T_ab and T_ba (vertical format)
    T = (diag_pre_multiply(rep_vector(s_T, 2), L_dyad) * z_T)';
    
    // Rename dyadic random effects
    vector[N_dyad] T_ab = T[, 1]; // T_ab
    vector[N_dyad] T_ba = T[, 2]; // T_ba
  
  
  // Expected dyadic rates
  vector[N] m_ab;
  vector[N] m_ba;
  
  // Two parallel linear models
  m_ab = exp(D + G[ind_a] + R[ind_b] + T_ab[dyad]);
  m_ba = exp(D + G[ind_b] + R[ind_a] + T_ba[dyad]);
}

model{
  // Priors
    // Fixed effects
    D ~ normal(0, 1);
    b_G ~ normal(0, 1);
    
    // Varying individual effects
    to_vector(z_K) ~ normal(0, 1);
    s_G ~ exponential(1);
    s_R ~ exponential(1);
    L_ind ~ lkj_corr_cholesky(2);
    
    // Varying dyadic effects
    to_vector(z_T) ~ normal(0, 1);
    s_T ~ exponential(1);
    L_dyad ~ lkj_corr_cholesky(2);
  
  // Likelihood
  y_ab ~ poisson(m_ab);
  y_ba ~ poisson(m_ba);
}

We run the statistical models:

# Run model
post_6_adj <- m_6_adj$sample(
  data = d6,
  iter_warmup = 1000,
  iter_sampling = 2000,
  chains = 4,
  parallel_chains = 4
)

post_6_adj %>% tidy_draws() %>% select(b_G) %>% pull() %>%
  saveRDS("./fitted_models/m_6_adj.rds")

# Run model
post_6_unadj <- m_6_unadj$sample(
  data = d6,
  iter_warmup = 1000,
  iter_sampling = 2000,
  chains = 4,
  parallel_chains = 4
)

post_6_unadj %>% tidy_draws() %>% select(b_G) %>% pull() %>%
  saveRDS("./fitted_models/m_6_unadj.rds")

And plot the marginal posterior distribution of \(\beta_{\gamma}\), for the two statistical models (adjusted by \(S_{[a, b]}\) at the bottom, and non-adjusted at the top):

Show code:

tibble(
  adj = readRDS("./fitted_models/m_6_adj.rds"),
  unadj = readRDS("./fitted_models/m_6_unadj.rds")
) %>%
  gather(param, value, 1:2) %>%
  ggplot(aes(x = value, y = param)) +
    
    # Theme and layout
    theme_bw() +
    labs(x = "", y = "", color = 'Percentile Interval') +
    theme(
      legend.position = "none",
      axis.text.y = element_text(size = 6.5),
      axis.text.x = element_text(vjust = -1.5),
      axis.ticks.y = element_blank(),
      panel.grid = element_blank(),
      panel.grid.major.y = element_line(color = "#c2c2b0",
                                        size = 0.3,
                                        linetype = "dotted"),
      strip.background = element_rect(fill = "white", color = "white")
    ) +
    geom_vline(xintercept = 0.5, color = "#b8b5ab", linewidth = 0.3, linetype = "dotted") + 
  
  # Slab
  stat_slab(
      slab_color = NA,
      fill = "#a19987",
      scale = 0.5,
      normalize = "groups",
      slab_alpha = 0.3,
      density = "bounded",
      trim = FALSE
    ) +
    
    # Contour of slab
    stat_slab(
      slab_color = "#363533",
      fill = NA,
      slab_linewidth = 0.1,
      scale = 0.5,
      normalize = "groups",
      slab_alpha = 0.75,
      density = "bounded",
      trim = FALSE
    ) +
      # Target values
      geom_point(data = tibble(value = c(0.5, 0.5),
                               param = c("unadj", "adj")),
                 pch = 21,
                 colour = "white",
                 fill = "#42747f",
                 size = 3)

We observe that the model that conditions on \(S_{|a,b|}\) (adj, bottom) recovers the true effect (in blue), whereas the model that does not condition on \(S_{|a,b|}\) (unadj, bottom) produces a biased estimate.